Linear Matrix Inequalities in Control [控制中的线性矩阵不等式]

is equivalent to the single LMI 1 F0 0 .. . + 0
m F0
n
xk
1 Fk
0 .. .
m Fk
≺ 0.
k=1
0
Assuming single LMI constraint causes no loss of generality. Good solvers exploit diagonal structure for computational efficiency!
8/23
Carsten Scherer
Siep Weiland
Truss Topology Design
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9/23
Carsten Scherer
Siep Weiland
Example: Truss Topology Design
1/23
Introduction
• Overview: What can be expected from LMI techniques? • What are LMI’s and what are they good for? • Example: Truss topology design • Software • Some aspects of linear algebra
Linear Matrix Inequalities in Control
Carsten Scherer Delft Center for Systems and Control (DCSC) Delft University of Technology The Netherlands Siep Weiland Department of Electrical Engineering Eindhoven University of Technology The Netherlands
7/23
Carsten Scherer
Siep Weiland
What are they good for?
• Many engineering optimization problem can be easily translated into LMI problems. • Various computationally difficult optimization problems can be effectively approximated by LMI problems. • In practice description of data is affected by uncertainty. Robust optimization problems can be either translated or approximated by standard LMI problems. Essential topic of this course How to translate/approximate a given (uncertain) optimization problem into/by an LMI problem?
• Connect nodes with N bars of length l = col(l1 , . . . , lN ) (fixed) and cross-sections s = col(s1 , . . . , sN ) (to-be-designed). • Impose bounds ak ≤ sk ≤ bk on cross-section and lT s ≤ v on total volume (weight). Abbreviate a = col(a1 , . . . , aN ), b = col(b1 , . . . , bN ). • If applying external forces f = col(f1 , . . . , fM ) (fixed) on nodes the construction reacts with the node displacement d = col(d1 , . . . , dM ). Mechanical model: A(s)d = f where A(s) is the stiffness matrix which depends linearly on s and has to be positive definite. • Goal is to maximize stiffness what amounts to minimizing the elastic stored energy f T d.
3/23
Carsten Scherer
Siep Weiland
Major Goals for Optimization and Control
• Distinguish easy from difficult problems: Convexity is key. • What are the consequences of convexity in optimization? • What is robust optimization? • How can we check robust stability by convex optimization? • Which performance measures can be incorporated? • How can controller synthesis be convexified? • What are the limits for the synthesis of robust controllers? • How can we perform systematic gain-scheduling?
Carsten Scherer
B, A
B, A
B defined/characterized analogously.
6/23
Siep Weiland
Observation: System of LMI’s is LMI
The system of m individual LMI’s
1 1 1 + x1 F1 F0 + · · · + xn Fn ≺0 . . . m m m F0 + x1 F1 + · · · + xn Fn ≺0
2/23
Carsten SchererSiep Weiland
Merging Control and Optimization
In contrast to classical control, H∞ -synthesis allows to design controllers in an optimal fashion. However the H∞ -paradigm is restricted: • Performance spec in terms of complete closed-loop transfer-matrix. Sometimes only particular channels are relevant. • One measure of performance with clear interpretation in frequency domain. Often particular time-domain specs have to be imposed. • No incorporation of structured time-varying/nonlinear uncertainties. • Can only design LTI controllers. View controller as decision variable of optimization problem. Desired specifications are constraints on controlled closed-loop system.
10/23
Carsten Scherer
Siep Weiland
Example: Truss Topology Design
Find s ∈ RN which maximizes f T d subject to the constraints A(s) Features • Data: Scalar v , vectors f , a, b, l, and symmetric matrices A1 , . . . , AN which define the linear mapping A(s) = A1 s1 + · · · + AN sN . • Decision variables: Vectors s and d. • Objective function: d → f T d which happens to be linear. • Constraints: Semi-definite constraint A(s) 0, nonlinear equality constraint A(s)d = f , and linear inequality constraints lT s ≤ v , a ≤ s ≤ b. Latter interpreted elementwise!
5/23
Carsten Scherer
Siep Weiland
Recap
For a real or complex matrix A the inequality A ≺ 0 means that A is Hermitian and negative definite. ¯T . If A is real then this • A is defined to be Hermitian if A = A∗ = A amounts to A = AT and A is called symmetric. Set of n × n Hermitian and symmetric matrices: Hn and Sn . All eigenvalues of Hermitian matrices are real. • Suppose A is Hermitian. By definition A is negative definite if x∗ Ax < 0 for all complex vectors x = 0. A is negative definite iff all its eigenvalues are negative. • A ≺ B, A
4/23
Carsten Scherer
Siep Weiland
Linear Matrix Inequalities (LMIs)
An LMI is an inequality of the form F0 + x1 F1 + · · · + xn Fn ≺ 0 where F0 , F1 , . . . , Fn are real symmetric matrices and x1 , . . . , xn are real scalar unknowns. LMI feasibility problem: Test whether there exist x1 , . . . , xn that render the LMI satisfied. LMI optimization problem: Minimize c1 x1 + · · · + cn xn over all x1 , . . . , xn that satisfy the LMI. Only simple cases can be treated analytically → Numerical techniques.
11/23
0, A(s)d = f, lT s ≤ v, a ≤ s ≤ b.